Resistivity is one of the most important parameters measured in wells for hydrocarbon exploration and production. For many years, conventional induction tools have been built with coils that have magnetic moment along the tool axis (co-axial dipole) that is mainly sensitive to the horizontal resistivity when the formation is horizontal and the well is vertical. Different designs of co-axial multi-coils array induction has appeared in the literature for over 30 years. Although theoretical work on resistivity anisotropy measurements started in 1950's (See Kunz and Moran, “Some effects of formation anisotropy on resistivity measurements in boreholes,” Geophysics 23, 770-794 (1958)), and the detailed theoretical derivation on the magnetic moment perpendicular to the tool axis (co-planer dipole) was also published in 1979 (Moran and Gianzero, “Effects of formation anisotropy on resistivity logging measurements,” Geophysics 44, 1266-1286 (1979)), the multi-component/tri-axial induction tool was not introduced into commercial service until the first decade of the 21st century. First, Baker Atlas commercialized its multi-component tool-3DEX (Schon et. al., “Aspects of Multicomponent resistivity data and macroscopic resistivity anisotropy,” SPE 62909 (2000)), and Schlumberger introduced their version of a tri-axial induction tool-AIT-Z (Rosthal et al., “Field test results of an experimental fully tri-axial induction tool”, SPWLA 44th Annual Symposium, Paper QQ (2003)); and Barber et. al., “Determining formation resistivity anisotropy in the presence of invasion”, SPE 90526 (2004)). However, these multi-component/tri-axial component tools have been mainly marketed as a thin-bed, low-resistivity-pay tool to invert horizontal and vertical resistivity Rh, and Rv, to be used with assumptions of vertical transverse isotropy (“VTI”) symmetry for (for example) thinly laminated formation, or Horizontal Transverse Isotropy (HTI) symmetry for vertical fractures (Rabinovich et al., “Determination of fracture orientation and length using multi-component and multi-array induction data,” U.S. Patent Application No. 2005/0256645 (2005)). Within the framework of a transverse isotropic model, Rv is assumed to be greater than Rh. (The tool used is designed mainly as a thin-bed tool for laminated formations. Then, in thinly laminated sand-shale sequences with high-low resistivity, the series combination of resistances (Rv) must be greater than the parallel combination (Rh).) Hence, when the inverted Rh is greater than Rv, one solution that has been used to reconcile this conflict is to force the horizontal resistivity to be equal to vertical resistivity, Rh=Rv, i.e., isotropy.
FIG. 1 illustrates a case where the inverted Rh is forced to equal Rv. Record 11 displays the recorded voltage measurements made with a multi-component/tri-axial induction tool. The dashed line is the Vxx data, the dotted line is Vyy and the solid line is Vzz. The different layers are illustrated at 12 as a function of depth. Shale layers are denoted by reference number 13; 14 is a thin-bed zone; and 15 is a layer with crossed beds or fractured beds. In the shale and thin-bed formations, Vxx=Vyy, i.e., these two orthogonal measurements match when rotated to be aligned, and in these formations the consequent inversion for Rh works as can be seen from record 16 where the dotted line represents Rh and the solid line Rv. But in the cross-bed or fractured bed formation 15, there are physical reasons why Vxx≠Vyy≠Vzz after azimuthal and dip rotations, and it can be seen in record 16 that the inversion method has had to force Rh and Rv to be equal in formation 15. Such problems can arise, for example, in the Piceance Basin where tight gas sandstone often have natural vertical fractures. Such results indicate that the Rv and Rh inversion is problematic when one cannot rotate the measured 9-component data to match the two orthogonal magnetic moments in the bedding plane, e.g., Hxx≠Hyy (FIG. 1). Record 17 is a corresponding gamma ray log.
As shown in the schematics of FIGS. 2A and 2B, the nine components (σij) of the conductivity tensor (conductivity and resistivity are mutually reciprocal quantities) measured by three pairs of the orthogonally orientated magnetic dipoles can be rotated through inversion to find σh, σv, within the framework of a VTI or HTI model. All other non-diagonal terms in the conductivity matrix should vanish when borehole, tool eccentricity, and other effects (such as invasion) are corrected, or become small enough to be ignored. In case the formation has VTI or uniaxial anisotropy, the relationship between the tensor conductivity in bedding coordinate and tri-axial induction measurements in borehole coordinates can be expressed by a rotation matrix (R), in the form {right arrow over (σ)}′={right arrow over (R)}T·{right arrow over (σ)}·{right arrow over (R)}, i.e.,
                                                        σ              →                        ij            ′                    =                                                                                          R                    →                                    T                                ·                                  [                                                                                                              σ                          xx                                                                                                                      σ                          xy                                                                                                                      σ                          xz                                                                                                                                                              σ                          yx                                                                                                                      σ                          yy                                                                                                                      σ                          yz                                                                                                                                                              σ                          zx                                                                                                                      σ                          zy                                                                                                                      σ                          zz                                                                                                      ]                                ·                                  R                  →                                            ⇒                              [                                                                                                    σ                        xx                                                                                    0                                                              0                                                                                                  0                                                                                      σ                        yy                                                                                    0                                                                                                  0                                                              0                                                                                      σ                        zz                                                                                            ]                                      ⁢                                                  ⁢                                                  =                          [                                                                                          σ                      h                                                                            0                                                        0                                                                                        0                                                                              σ                      h                                                                            0                                                                                        0                                                        0                                                                              σ                      v                                                                                  ]                                      ⁢                                  ⁢        where        ⁢                                  ⁢                              R            →                    =                      [                                                                                cos                    ⁢                                                                                  ⁢                    αcosβ                                                                                        cos                    ⁢                                                                                  ⁢                    α                    ⁢                                                                                  ⁢                    sin                    ⁢                                                                                  ⁢                    β                                                                                                              -                      sin                                        ⁢                                                                                  ⁢                    α                                                                                                                                          -                      sin                                        ⁢                                                                                  ⁢                    β                                                                                        cos                    ⁢                                                                                  ⁢                    β                                                                    0                                                                                                  sin                    ⁢                                                                                  ⁢                    αcos                    ⁢                                                                                  ⁢                    β                                                                                        sin                    ⁢                                                                                  ⁢                    αsin                    ⁢                                                                                  ⁢                    β                                                                                        cos                    ⁢                                                                                  ⁢                    α                                                                        ]                                              (        1        )            In many cases, the above expression cannot be realized by rotation and inversion because the formation simply has higher anisotropic symmetry than VTI system. FIG. 3A shows an aeolian outcrop section with complicated bedding planes (indicated by solid lines 41), cross-bedding planes (indicated by dotted lines 42), and fault/fracture planes (indicated by broken lines 43). In such a more general anisotropic case, the process expressed in Equation (1) can never be realized by a matrix rotation and inversion process such as is shown in FIG. 1, where the inversion process is limited to an assumption of VTI, or HTI, or in general, uniaxial anisotropic symmetry with arbitrary plane. Note that the thickness of crossbed varies in the range of meters in the large view of FIG. 3A (see the cartoon people in the lower right for scale), but also includes fine layers in the range of mm to cm as shown in the exploded view of FIG. 4B (ball-point pen shown as scale indicator). In such more general anisotropic cases, assumptions of the TI symmetry for tri-axial induction measurements and inversion will break down, and dip, azimuth, and resistivity anisotropy will be different if they are inverted from the data taken from transmitter-receiver pairs with different spacing from multi-spacing tools.
What is needed is an inversion method that can use all nine components that can be measured by tri-axial induction tools, plus borehole azimuth and deviation data, to address these issues and solve for the tri-axial induction response in an arbitrary anisotropic formation due to the non-orthogonal bedding plane and fracture plane (or cross bedding plane). The present invention provides such a method.